1. Minimum Spanning Trees and Shortest Paths
Kruskal's Algorithm
Prim's Algorithm
Shortest Paths
April 04, 2018
Cinda Heeren / Geoffrey Tien

2. Kruskal's algorithm
Data types for implementation
KruskalsAlgorithm() {
set 𝐸′ = ø
while ( 𝐸′ ≠ 𝑉 −1)
Find minimum weight edge 𝑒∉𝐸′ such that 𝐸′ 𝑒 does not contain cycles
Add 𝑒 to 𝐸′ }
We need ADTs that support our required operations efficiently
How do we find the minimum weight edge?
Priority queue!
How can we check for cycles and perform union?
Disjoint sets!
April 04, 2018
Cinda Heeren / Geoffrey Tien

3. Cinda Heeren / Geoffrey Tien
Kruskal's algorithm
Example A B 2
prQ
MST weight: 38 6 3
𝐴,𝐵 2
𝐹,𝐺 10 16 7 C 8 17
𝐵,𝐶 3
𝐹,𝐻 11 D E 13 12 F 5 13
𝐺,𝐻 4
𝐷,𝐹 12 9 16 10 11 A
𝐸,𝐹 5
𝐵,𝐺 13 G H 4
𝐴,𝐶 6
𝐶,𝐹 13 E B C D G F H
𝐵,𝐸 7
𝐴,𝐷 16
prQ cost
heap
ordered array
build
removeMin
𝐶,𝐷 8
𝐸,𝐻 16
Overall cost?
𝐷,𝐺 9
𝐶,𝐸 17
Note: only one edge direction listed in prQ for compactness in this slide
Note that no insertions performed after build
April 04, 2018
Cinda Heeren / Geoffrey Tien

4. A slight tangent – maze construction
What makes a good maze?
A bunch of adjacent rooms
Each room is a vertex
Open wall between rooms form edge
Unpredictable, not easily solved
Highly branching, many dead ends
Just enough walls to get from any room to any other room
Especially start and finish in
out
April 04, 2018
Cinda Heeren / Geoffrey Tien

5. Maze under construction
So far, a number of walls have been knocked down, while others remain
Now we consider the wall between rooms A and B
Should we knock it down?
If A and B are otherwise connected
If A and B are not otherwise connected A B
Algorithm:
While edges remain in 𝐸
Remove a random edge 𝑒= 𝑢,𝑣 from 𝐸
If 𝑢 and 𝑣 have not been connected
Add 𝑒 to 𝐸′
Mark 𝑢 and 𝑣 as connected
This is a lot like Kruskal's algorithm!
Solve it using disjoint sets and random edge selection
April 04, 2018
Cinda Heeren / Geoffrey Tien

6. Recall: BFS spanning tree
Starting from vertex A A B 2 A
(B,2), (C,6), (D,16) B
(A,2), (C,3), (E,7), (G,13) C
(A,6), (B,3), (D,8), (E,17), (F,13) D
(A,16), (C,8), (F,12), (G,9) E
(B,7), (C,17), (F,5), (H,16) F
(C,13), (D,12), (E,5), (G,10), (H,11) G
(D,9), (F,10), (H,4) H
(E,16), (F,11), (G,4) B C D 6 3 E G 16 7 C F 8 17 D E 13 12 F 5 H 13 9 16 10 11 G H 4
Queue: A B C D E G F H
Identified: T T T T T T T T A B C D E F G H
What if we use a priority queue (with neighbours' edge weights) instead of an ordinary queue?
April 04, 2018
Cinda Heeren / Geoffrey Tien

7. Cinda Heeren / Geoffrey Tien
Prim's algorithm
Greedy algorithm
Builds a spanning tree from initially one vertex.
Repeatedly chooses the minimum-weight edge from a vertex in the tree, to a vertex outside the tree – adds that vertex to the tree
PrimsAlgorithm(v) {
mark v as visited, add v to spanning tree
while (graph has unvisited vertices)
Find least cost edge (w, u) from a visited vertex w to unvisited vertex u
Mark u as visited
Add vertex u and edge (w, u) to the minimum spanning tree }
April 04, 2018
Cinda Heeren / Geoffrey Tien

8. Cinda Heeren / Geoffrey Tien
Prim's algorithm A B 2 B A
(B,2), (C,6), (D,16) B
(A,2), (C,3), (E,7), (G,13) C
(A,6), (B,3), (D,8), (E,17), (F,13) D
(A,16), (C,8), (F,12), (G,9) E
(B,7), (C,17), (F,5), (H,16) F
(C,13), (D,12), (E,5), (G,10), (H,11) G
(D,9), (F,10), (H,4) H
(E,16), (F,11), (G,4) 6 3 C E 16 7 C D 8 17 D E G 13 12 F 5 F 13 9 16 10 11 H G H 4
(A,B,2)
(A,C,6)
(A,D,16)
(B,C,3)
(B,E,7)
prQ:
Visited: T T T T T T T T
(B,G,13)
(C,D,8)
(C,E,17)
(C,F,13)
(E,F,5) A B C D E F G H
(E,H,16)
(F,D,12)
(F,G,10)
(F,H,11)
(D,G,9)
(G,H,4)
All vertices visited
MST weight: 38
April 04, 2018
Cinda Heeren / Geoffrey Tien

9. Cinda Heeren / Geoffrey Tien
Prim's algorithm
Complexity
Unlike Kruskal's algorithm, we will intersperse insertion and removal operations to the priority queue
Maximum number of insertions into the priority queue?
Assuming heap implementation of prQ
𝐸 in the worst case, then total cost of all insertions 𝑂 𝐸 log 𝐸
For dense graphs, 𝐸 ∈𝑂 𝑉 2 , then log 𝐸 ∈𝑂 2∙ log 𝑉 ∈𝑂 log 𝑉
Thus the complexity of Prim's algorithm is 𝑂 𝐸 log 𝑉
Actually we can describe Kruskal's algorithm the same way
April 04, 2018
Cinda Heeren / Geoffrey Tien

10. Single-source shortest path
Given a graph 𝐺= 𝑉,𝐸 and a vertex 𝑠∈𝑉, find the shortest path from 𝑠 to every vertex in 𝑉
Variations
Weighted vs unweighted
Cyclic vs acyclic
Positive weights only vs negative weights allowed
Multiple weight types to optimize
April 04, 2018
Cinda Heeren / Geoffrey Tien

11. Single-Source Shortest Path
Un/directed, weighted graphs, no negative cycles
What is the least cost path from one vertex to another?
For weighted graphs, this is the path that has the smallest sum of its edge weights A 1 4 B C 5 3 2 E 1 5 D 8 1 F 2 G
The shortest path between B and G is:
B-D-E-F-G (7) and not
B-G (8)
April 04, 2018
Cinda Heeren / Geoffrey Tien

12. Cinda Heeren / Geoffrey Tien
Dijkstra's Algorithm
Classic algorithm for solving shortest path in weighted graphs without negative weights
A greedy algorithm
Best local choice is made at each step, without considering future consequences
Intuition:
Shortest path from source vertex to itself is 0
Cost of going to adjacent nodes is at most edge weights
Cheapest of these must be shortest path to that node
Update paths for new node and continue picking shortest path
April 04, 2018
Cinda Heeren / Geoffrey Tien

13. Cinda Heeren / Geoffrey Tien
Dijkstra's Algorithm
Initialize the cost of reaching each vertex to ∞
Initialize the cost of the source to 0
While there are unvisited vertices left in the graph
Select the unvisited vertex with the lowest cost: 𝑢
Mark 𝑢 as visited, and note the vertex 𝑣 which was used to reach 𝑢
For each vertex 𝑤 which is adjacent to 𝑢
𝑤's cost = min(𝑤's old cost, 𝑢's cost + cost of (𝑢,𝑤))
And note the "parent" vertex which can be used to reach 𝑤 with the lowest cost
April 04, 2018
Cinda Heeren / Geoffrey Tien

14. Dijkstra's Algorithm Example: directed graph A B C D E F G H 2 4 7 9 18 10 3
Source node: C
Priority queue:
Vertex
Cost
Parent A B C D E F G H
Visited
Cost
Parent
Note that we need access to arbitrary elements in this prQ in order to update
April 04, 2018
Cinda Heeren / Geoffrey Tien

15. Cinda Heeren / Geoffrey Tien
Exercise
Given the following undirected, weighted graph, use Kruskal's algorithm and Prim's algorithm to find a minimum spanning tree and write its weight. Is the tree you found the unique MST for this graph? A 12 B 8 9 9 C 10 8 5 5 D 6 E F 14 G 3 4 H 12 7 4 9 I 7 J 8 K
April 04, 2018
Cinda Heeren / Geoffrey Tien

16. Readings for this lesson
Carrano & Henry
Chapter (Minimum spanning trees)
Chapter (Shortest paths)
April 04, 2018
Cinda Heeren / Geoffrey Tien